Hartshorne's Conjectures about Algebraic Bundles? Algorithmic Preliminaries 12 5. You remember correctly, here's the paper: At the end of her talk at the Hyderabad Congress, Claire Voisin was asked by someone whether she believed in the Hodge conjecture. Books 2 1.2. Every algebraic action of $\mathbb{C}^*$ on $\mathbb{C}^n$ is linear in some coordinates of $\mathbb{C}^n$. If it did, then this problem would no longer be an open problem. * Standard conjectures on algebraic cycles (though these are not so urgent since Deligne proved the Weil conjectures). A rational cuspidal curve in $\mathbb{P}^2$ is rectifiable, i.e. In connection to vector bundles over $\mathbb{P}^n$, Hartshorne's paper from 1979 provides a list of open problems. Use MathJax to format equations. Open problems in Birational Geometry, after BCHM. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Topological Preliminaries 22 6. Open problems in Birational Geometry, after BCHM, Open Problems in Algebraic Topology and Homotopy Theory. Swapping out our Syntax Highlighter, Responding to the Lavender Letter and commitments moving forward, The most outrageous (or ridiculous) conjectures in mathematics. Are there other problems that I missed? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Do you mean open problems mentioned in Hartshorne's book or conjectures that he made? The ArXiv gets "proofs" of the Jacobian conjecture (I.3.19d) reasonably often. Thanks for contributing an answer to MathOverflow! MO questions like the rest of us need luck. Making statements based on opinion; back them up with references or personal experience. Her answer was equivocal, if memory serves me right. @Matan Fattal This is the list of open problems, Open problems in “Algebraic geometry by robin hartshorne”, sciencedirect.com/science/article/pii/0040938379900302, Goodbye, Prettify. Use MathJax to format equations. Coolidge-Nagata Conjecture. The paper is "Algebraic vector bundles on projective spaces: A problem list" Topology, 18:117–128, 1979. MathOverflow is a question and answer site for professional mathematicians. I've lost track of the number of false claims regarding this on the arxiv and elsewhere. Open for $m>2$. Updates to Stanley's 1999 survey of positivity problems in algebraic combinatorics? This question was lucky enough that Richard Borcherds offered a very nice answer and potentially there will be further answers that we can enjoy and ultimately this will be a useful source. 3 2.1. In that process, the search for finding the “true” nature of the problem at hand is the impetus for our thoughts. Contents 1. MathJax reference. problems mentioned in hartshornes book (the ** excercises). Lecture 2: Prestacks 8 3.1. One of them is atonishingly simple but still completely open : Let $E$ a rank $2$ vector bundle on $\mathbb{P}^n$, with $n \geq 7$. Usually recent ICM talks, survey articles in the bulletin, and recently published advanced textbooks are good places to start for this kind of thing. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. To learn more, see our tips on writing great answers. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. In a stronger form Hartshorne's conjecture says that any codimension $>\frac{2}{3}n$ subvariety of $\mathbf{P}^n_{k},k$ an algebraically closed field is a complete intersection. Is there a precise relationship between the goals of moduli theory and the minimal model program? My opinion was that just like in "real world mathematics" (and science) attracting good answers is a merit of a question. There are examples of indecomposable rank $2$ vector bundles on $\mathbb{P}^5$ in characteristic $2$ due to Tango and Kumar-Peterson-Rao (independently). Let me mention a couple of problems related to vector bundles on projective spaces. We've had many discussions over at meta about whether a sufficient condition to be a good question is that it generates good answers. Exercises 1: categorical preliminaries 6 3. If $X\times \mathbb{C}\cong \mathbb{C}^{m+1}$ then $X\cong \mathbb{C}^m$. Lecture Notes 2 1.3. Farkas proved that $\overline{M}_g$ is of general type for $g = 22$. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. For a good introduction to the subject, allow me to recommend the book, I believe the Coolidge-Nagata conjecture is now known, see, Well, Y.T Siu has recently claimed he had proved the abundance conjecture (in a version stating that the Kodaira dimension equals the numerical Kodaira dimension); here's the paper : arxiv.org/abs/0912.0576, For some additional discussion of Siu's work, see the recent question. Publisher: International Press of Boston, Inc. The Tate conjecture: Let $k$ be a finitely generated field, $X/k$ a smooth projective geometrically integral variety and $\ell$ invertible in $k$. Why don't you read some of the literature on these topics to find out? Complex vector bundles that are not holomorphic, Algebraic vector bundles on projective spaces: A problem list, http://www.math.harvard.edu/~chaoli/doc/TateConjecture.html, Goodbye, Prettify.

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